Infinite cycles in the interchange process in five dimensions

TL;DR

This paper proves the existence of infinite cycles in the interchange process on high-dimensional lattices for large times, confirming a long-standing conjecture and introducing a novel self-interacting random walk analysis.

Contribution

It establishes the presence of infinite cycles in the interchange process on  \, orall d \u2265 5, 0 for large eta, and introduces a new analytical approach using a cyclic time random walk.

Findings

Infinite cycles exist in  0 for large eta.

The cyclic time random walk is diffusive and couples with Brownian motion.

A local escape property is key to the proof.

Abstract

In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi_\beta:V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta $, establishing a conjecture of B\'alint T\'oth from 1993 in these dimensions. In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.